Maxime J Jacquet 1, 2, ∗ and Friedrich K¨onig 3, †
1
Laboratoire Kastler Brossel, Sorbonne Universit´e, CNRS, ENS-Universit´e PSL, Coll`ege de France, Paris 75005, France
2
Faculty of Physics, University of Vienna, Boltzmanngasse 5, Vienna A-1090, Austria.
3
School of Physics and Astronomy, SUPA, University of St. Andrews, North Haugh, St. Andrews, KY16 9SS, United Kingdom
We consider a moving refractive index perturbation in an optical medium as an optical analogue to waves under the influence of gravity. We describe the dielectric medium by the Lagrangian of the Hopfield model. We supplement the field theory in curved spacetime for this model to solve the scattering problem for all modes and frequencies analytically. Because of dispersion, the kinematic scenario of the field modes may contain optical event horizons for some frequencies. We calculate the spectra of spontaneous emission in the frame co-moving with the perturbation and in the laboratory frame. We also calculate the spectrally-resolved photon number correlations in either frame. The emitted multimode field comes in different types depending on the presence of horizons. We show that these types are robust against changes in the system parameters and thus are genuine features of optical and non-optical analogues. These methods and findings pave the way to new observations of analogue gravity in dispersive systems.
I. INTRODUCTION
A number of classical and semi-classical features of grav- ity can be reproduced in the laboratory: it is possible to create effectively curved spacetimes for waves in media [1] and, in particular, event horizons. These horizons scatter waves and are predicted to spontaneously emit quanta by the Hawking effect [2, 3]. Recently, experiments in many different ‘fluid’
systems, such as Bose-Einstein condensates [4, 5], water [6–
11] and polariton microcavities [12] have demonstrated hori- zons and studied the behaviour of waves in their vicinity. New developments in quantum fluids of light [13–15] may soon en- able novel analogue gravity experiments.
As in the astrophysical case [3], the paired spontaneous emission in analogues results from mixing of field modes of positive and negative Klein-Gordon norm at the horizon [1].
Thus, pairs from horizons have been extensively studied for fluid systems, in which their nonseparability in various dis- persive regimes has been investigated [16–21], and are con- sidered an unmistakable signature of the Hawking effect [22–
27].
It is also possible to create effectively curved spacetimes for light in dispersive media. As an optical pulse moves through the dispersive medium, the pulse intensity locally raises the refractive index n of the medium by the optical Kerr effect, creating a moving refractive index front (RIF). Light under the pulse is slowed by the increased index, i.e., light of some frequencies will be slowed below the pulse speed and captured into the RIF. This is in analogy with the kinematics of waves at a black-hole event horizon [28–37]. At other frequencies, however, the kinematics of light waves are different [38, 39], recreating wave motion at a white-hole event horizon or two- way motion towards and away from the RIF. Emission in all these kinematic scenarios occur simultaneously.
∗
maxime.jacquet@lkb.upmc.fr
†
fewk@st-andrews.ac.uk
Figure 1. Schematic of the Hawking effect at the refractive index front (RIF). The RIF separates two homogeneous index regions on the left (x < 0) and on the right (x > 0) of a dielectric boundary of height δn. The stationary process is shown in the frame co-moving with the RIF in the dispersive dielectric. Incoming vacuum modes (grey) scatter on the boundary at x = 0, which yields the spontan- eous emission of entangled photons pairs.
The RIF may be time independent in the moving frame: the system is stationary and can be described efficiently by the scattering matrix. Possible analytical methods to arrive at the scattering matrix broadly fall under two categories, depending on the profile of the RIF considered: for smooth RIFs [40, 41], the JWKB approximation is used to calculate the asymptotic states (far from the RIF) that are matched with the near-RIF expansion (which is instead treated in Fourier space), whereas no approximations are needed in the case of a step-like RIF [42]. Although the step is physically hard to implement, be- cause the particles are emitted from a single point, this simple optical system allows us to contrast the quantum emission in the different kinematic scenarios highlighted above. The out- put fields will be further modified by gradual horizons, two- horizon interactions and other nonlinear effects such as optical Cerenkov radiation. Studying the RIF allows us to identify ˇ which quantum effects are connected with a single horizon.
The approach of [42] needs to be comprehensively general-
ised to account for the possible change of kinematic scenarios,
and to include a treatment of evanescent waves in these scen-
arios as well as a derivation of the full scattering matrix. From
the scattering matrix follow the output quantum fields for all modes. In particular, we are interested in key output observ- ables — such as photon-number spectra and correlations — which are of principal importance in any theoretical or ex- perimental study of analogue gravity. The calculation of the photon-number correlations is challenging because it requires accounting for the dispersion and the bandwidth of the modes.
In this paper, we supplement and extend the theoretical de- scription of the spontaneous emission at a step-like RIF of [42], as shown in Fig.1. We present an analytical method to calculate the scattering matrix in all kinematic scenarios in a dispersive dielectric featuring intra- and inter-branch scatter- ing. Moreover, we demonstrate the calculation of key quant- ities measurable in the medium rest frame, viz. the photon emission spectral density and the spectrally-resolved photon- number correlations, as they are recorded with a variable bandwidth detector. We find pertinent features when modi- fying the parameters of the system.
In section II, we review all possible kinematic scenarios for waves at the RIF and explain their relevance in the anlogy with gravity. Section III derives the photon flux in differ- ent frames as well as the spectral correlations detected with variable bandwidth detectors in terms of the scattering mat- rix. Section IV derives the scattering matrix for all modes and frequencies from the field matching conditions at the RIF. In section V we use our method to compute observables in bulk fused silica. We calculate key observable quantities, namely the spectral density of spontaneous emission and the spectral correlations, in the rest frame of the medium and discuss their dependence on the RIF height and velocity. We also briefly discuss desired medium properties.
II. KINEMATIC SCENARIOS
Our study is based on the consideration of the step-like geo- metry of a RIF that propagates at constant speed u in the posit- ive X-direction in the laboratory frame. This RIF is illustrated in Figure 1 in co-moving frame coordinates x and t. In Ap- pendix A we review and supplement the field theory for light in an inhomogeneous dispersive dielectric. There we find the modes of the homogeneous medium and construct the ‘global modes’ (GMs) of the inhomogeneous system (including the step). Finally we quantise the field which is represented by annihilation and creation operators of the GMs. In this sec- tion, we review the kinematics of light waves at different fre- quencies [38] and explain the analogy with gravity.
The RIF separates two regions of homogeneous refractive index whose dispersion is modelled by (cf. App A)
c 2 k 2 = ω 2 +
3
X
i=1
4πκ i γ 2 (ω + uk) 2 1 − γ
2(ω+uk) Ω
2 2i
, (1)
with ω and k the frequency and wavenumber in the frame co- moving with the RIF at speed u (γ = [1−u 2 /c 2 ] −1/2 ). Ω i and κ i are the medium resonant frequencies and elastic constants.
The change in index, δn, at x = 0 between the two regions, is modelled by a change in κ i and Ω i in (1). As illustrated
0
0
0
0
=0
=0
=0
=0
b
Figure 2. Possible kinematic scenarios at the RIF. Time (left) and frequency (right) illustrations of propagating modes are shown for different comoving frequencies ω (a-d). The black (orange (light grey)) dispersion curve corresponds to the low (high) index region.
Single-frequency modes of ω are identified by intersections with the blue dashed line. Possible kinematic scenarios, analogous to changes of spacetime curvature: a, horizonless scenario (ω < ω
minL); b, white hole scenario (ω
minL< ω < ω
minR); c, horizonless scenario (ω
minR< ω < ω
maxL); d, black hole scenario (ω
maxL< ω <
ω
maxR).
in the dispersion diagrams of Fig.2, the step manifests itself as a change in the dispersion relation between the low (x >
0, black curve) and high (x < 0, orange (light grey) curve) refractive index regions.
The two parts of the optical branch, shown in Fig.2 for frequencies of relevance, represent negative- (positive-) norm modes on the left (right) in the diagram. Modes are marked with open circles. For all ω, there is one negative-norm mode (thin) and, in addition, either one or three positive- norm modes (thick). We refer to frequency intervals of ω with three positive norm modes as subluminal intervals (Sb- LIs): [ω minL , ω maxL ] and [ω minR , ω maxR ]. Inside a SbLI, one of the four mode solutions has a positive group velocity
∂ω
∂k in the moving frame, moving rightwards. We call this mode ‘mid-optical’ — ‘moL’ on the left, and ‘moR’ on the right of the boundary. Note that in no other mode may light propagate with positive group velocity. There is also the ‘low optical’ mode loL/R, the ‘upper optical’ mode uoL/R, and the
‘negative optical’ mode noL/R. Beyond the SbLIs, i.e. for
ω ≤ ω minL/R or ω ≥ ω maxL/R , only two propagating modes remain. Thus, in growing order of ω we find the following distinct kinematic scenarios at the step:
- Fig.2 a. Two optical propagating modes (noL/R, uoL/R) are present, with negative group velocities in the mov- ing frame on either side. Light in all modes may only move to the left.
- Fig.2 b. On the left of the interface, four optical propagating modes are present (noL, loL, moL, uoL) whilst only two modes (noR, uoR) are present on the right. Mode moL has positive group velocity, whereas all other modes have negative group velocity. Light can propagate into the boundary from the left, but cannot proceed further to the right. The interface acts as a white hole horizon to light.
- Fig.2 c. Four propagating modes (noL/R, loL/R, moL/R, uoL/R) exist on either side of the interface. Modes moL and moR have positive group velocity on the left and on the right of the RIF, respectively. The RIF is not a one-way door.
- Fig.2 d. On the left of the interface, two propagating modes are present (noL, uoL), whilst four optical modes are present on the right (noR, loR, moR, uoR). Light on the right can move in either direction, but on the left of the interface both modes have negative group velocity.
There is a one-way door from the right to the left: light experiences a black-hole horizon at the RIF.
For the largest ω, the kinematics return to the type of Fig.2 a.
Only in scenarios b and d is a subluminal region paired with a superluminal region, creating a horizon. This is in ana- logy to the superluminal space flow in the interior region of a black- or white hole and the subluminal flow outside [39].
The analogy to gravity is best understood when thinking of black hole emission by the Hawking effect: a positive norm mode — Hawking radiation — allows for energy to propagate away from the hole in the subluminal region, while its neg- ative norm partner falls inside the horizon. The black hole kinematics as illustrated in Fig.1 are realised in Fig.2 d: mode moR allows for light to escape the horizon to the right, like Hawking radiation, and mode noL is the infalling partner.
This analysis of kinematic scenarios is interesting because it shows that optical analogues may allow us to access fea- tures of gravity physics other than wave motion at the event horizon of black holes. The simple geometry of a localised step allowed us to identify the above kinematic scenarios. A continuous RIF would also exhibit one of these four scenarios at any one point x.
In all 4 kinematic scenarios, the motion of the RIF in the dispersive medium creates a non-adiabatic modification of the refractive index in time T , which corresponds to a time- dependance of the spacetime curvature. Thus, modes (even in the vacuum state) will scatter at the RIF. We use the S-matrix formalism to describe this mode coupling.
III. ANALYTICAL CALCULATION OF SPECTRAL DENSITIES AND CORRELATIONS
The aim of the field theory summarised in appendix A is to enable the calculation of all output properties in terms of the scattering matrix S. In [38], we calculated the output spectral density in all output modes. Paired production in these modes, which is at the heart of the Hawking effect, is best character- ised by the photon number variances and covariances. Here, we derive these quantities for an optical system for the first time.
In appendix A 2 we construct two basis sets of global modes (GMs) of the system: the in and out bases. We calculate the spontaneous emission in out modes, i.e., the in modes are in the vacuum state
0 in
. Because the system is in a stationary state, the photon flux for an out mode α is given by the integral over the frequency correlation [43]:
φ α (ω) = D
0 in | φ ˆ α (ω)|0 in E
=
∞
Z
0
dω 0 2π
0 in |ˆ a out α† (ω)ˆ a out α (ω 0 )|0 in .
(2)
The flux φ(ω) is the dimensionless number of photons per unit time δτ and unit bandwidth δω at ω in the moving frame.
We now replace the out-mode operators in the frequency correlations using the scattering transformation (A18) by in- operators, which act directly on the in vacuum state. Using the frequency correlation (E2) calculated in appendix E, we express the photon flux by the S-matrix
φ α (ω) = 1 2π
X
β / ∈{α}
|S αβ (ω)| 2 . (3)
{α} is the set of modes which have a positive (negative) norm if α is of positive (negative) norm. The photon flux results from the scattering of in modes into out modes of opposite sign of norm. The number of emitted photons depends solely on the scattering matrix S, bandwidth and interaction time.
The total photon number is obtained by integrating over all frequencies and time. To obtain the photon number N ˆ t α over a limited ‘detected’ frequency interval in this stationary mode conversion process, the field operators have to be constrained to this bandwidth. This is achieved via the spectral filter func- tion t(ω), which takes the value of 1 if the frequency is in the detected interval and 0 otherwise. Then
N ˆ t α = τ Z ∞
0
√ dω
2π t(ω)ˆ a out α (ω) † Z ∞
0
√ dω
2π t(ω)ˆ a out α (ω)
. (4) τ is the interaction time in the moving frame. From (4), one may calculate other important quantities, such as (3) and the photon-number correlations across the spectrum.
Correlations in photon number between modes α and α 0
are found by calculating the normally ordered covariance of
the photon numbers N ˆ 1 α and N ˆ 2 α
0on detectors 1 and 2:
cov( ˆ N 1 α , N ˆ 2 α
0) = τ 2
∞
Z Z Z Z
0
dω...dω 000
(2π) 2 t ∗ 1 (ω)t ∗ 2 (ω 0 )t 2 (ω 00 )t 1 (ω 000 )×
ˆ a α† (ω)ˆ a α
0† (ω 0 )ˆ a α
0(ω 00 )ˆ a α (ω 000 )
− N ˆ 1 α N ˆ 2
α
0. (5) In (5) the expectation value is again taken with respect
to the in vacuum state 0 in
and t 1 and t 2 are the filters for detectors 1 and 2, respectively. This defines the spec- tral intervals ∆ 1 and ∆ 2 over which these detectors collect photons. We calculate the fourth order moment of the field D
ˆ
a α† (ω)ˆ a α
0† (ω 0 )ˆ a α
00(ω 00 )ˆ a α
000(ω 000 ) E
in Appendix E. Insert- ing the result (E5) into (5) we find
cov( ˆ N 1 α , N ˆ 2 α
0) = τ 2π
2
∞
Z
0
∞
Z
0
dω dω 0
δ A X
β,β
0∈{α} /
S αβ ∗
0(ω)S α ∗
0β
0(ω)S α
0β (ω 0 )S αβ (ω 0 )t 1 (ω)t 2 (ω)t 2 (ω 0 )t 1 (ω 0 )
+ δ S X
β,β
0∈{α} /
S αβ ∗
0(ω)S α ∗
0β (ω 0 )S α
0β
0(ω)S αβ (ω 0 )t 1 (ω)t 2 (ω 0 )t 2 (ω)t 1 (ω 0 )
+ X
β / ∈{α}
X
β
0∈{α /
0}
|S αβ (ω)| 2 |S α
0β
0(ω 0 )| 2 |t 1 (ω)| 2 |t 2 (ω 0 )| 2
− N ˆ 1 α N ˆ 2
α
0.
(6)
Here S αβ is identical (complex conjugate) to S αβ if mode α is of positive (negative) norm. δ S , (δ A ) equals unity if the correlated modes α and α 0 have identical (opposite) norm and zero otherwise. Reverting from S back to S, we observe that in the δ A -term two S elements have to be conjugated and in the δ S -term either all or none. Since the result, as well as t i (ω), are real, the first two sums in (6) are equal. Hence we observe that the mutually exclusive and exhaustive δ S and δ A
let us combine the first two terms into one and the last two terms cancel due to (A18), (3) and (4):
cov( ˆ N 1 α , N ˆ 2 α
0) = τ 2π
2 X
β / ∈{α}
Z
∆
dω S αβ ∗ (ω)S α
0β (ω)
2
. (7)
∆ is the spectral overlap (moving frame frequencies) of the
two spectral intervals ∆ 1 and ∆ 2 of the two detectors. This result allows us to quantify the spectrally resolved photon number correlations of any stationary process in quantum op- tics. The correlations are contained in the scattering matrix and are dependent on the spectral overlap of the detectors re- garding the investigated mode.
We also calculate the variance var( ˆ N 1 α ) = h N ˆ 1 α N ˆ 1 α i − h N ˆ 1 α i 2 . Using the not normally ordered 4 th moment derived in (E6) of App. E in a not normally ordered expression equi- valent to (5), and with α = α 0 , we obtain
var( ˆ N 1 α ) = h N ˆ 1 α i
h N ˆ 1 α i + τ∆ 1 2π
. (8)
The photon-flux Pearson correlation coefficient between detectors 1 and 2, corresponding to modes α and α 0 . It is
C( ˆ N 1 α , N ˆ 2 α
0) ≡ cov( ˆ N 1 α , N ˆ 2 α
0) var( ˆ N 1 α ) var( ˆ N 2 α
0) 1/2
=
P
β / ∈{α}
R
∆
dω S αβ ∗ S α
0β
2
P
β / ∈{α}
R
∆
1dω|S αβ | 2
! P
β / ∈{α}
R
∆
1dω|S αβ | 2 + ∆ 1
! P
β / ∈{α
0}
R
∆
2dω|S α
0β | 2
! P
β / ∈{α
0}
R
∆
2dω|S α
0β | 2 + ∆ 2
! 1/2
= ∆ 2
∆ 1 ∆ 2
P
β / ∈{α}
S αβ ∗ S α
0β
2
P
β / ∈{α}
|S αβ | 2 P
β / ∈{α}
|S αβ | 2 + 1
! P
β / ∈{α
0}
|S α
0β | 2 P
β / ∈{α
0}
|S α
0β | 2 + 1
! 1/2 . (9)
In the last step we have assumed the scattering matrix to change little across a narrow detection bandwidth. The norm- alisation ensures that |C( ˆ N 1 α , N ˆ 2 α
0)| ≤ 1. Note that this also means that C is independent of the photon flux. The correla- tions generated by the scattering are entirely positive, indicat- ing their origin of entangled photon pair generation from the vacuum. From (9), we can easily extract the (narrow band- width) self-correlation with matched filters
C( ˆ N 1 α , N ˆ 1 α ) = h N ˆ 1 α i 2
var( ˆ N 1 α ) = 1 − var( ˆ N 1 α ) h N ˆ 1 α i τ∆ 2π
! −1
, (10)
which is a measure of the noise var( ˆ N 1 α ) in mode α relative to the Poisson noise variance of h N ˆ 1 α i τ∆ 2π . Using (8), we see that C increases from 0 (vacuum) and approaches 1 (maximum noise) for increasing photon numbers.
Furthermore, we calculate the second order correlation function g (2) αα
0= h: ˆ N
1αN ˆ
α0 2
:i
h N ˆ
1αih N ˆ
2α0i . Again assuming narrowband de- tection, we obtain:
g (2) αα
0= ∆ 2
∆ 1 ∆ 2
1 +
| P
β / ∈{α}
S αβ ∗ S α
0β | 2 P
β / ∈{α}
|S αβ | 2 P
β / ∈{α
0}
|S α
0β | 2
(11) as well as the single detector correlation
g αα (2) = 2. (12) Here we recover the expected result that the scattering induces correlated noise between the two detectors, both of which de- tect the statistics of chaotic light. These relations describe output observables in any process described by the scatter- ing matrix but are particularly tailored to analogue gravity in dispersive systems, in which context they appear for the first time.
The photon numbers and correlations up to this point are calculated in the moving frame where the system is station- ary. However, in an experiment, measurements are performed in the laboratory frame, so we now proceed to derive the flux there. Photon numbers and time-bandwidth products τ∆
are frame invariant. Therefore, re-interpreting τ and ∆ as the laboratory interaction time and detection bandwidth, the laboratory frame correlation coefficient of modes α and α 0 is identical to Eq.(9). All other relations containing photon number operators can be interpreted for the laboratory frame as well.
For light waves to reach the detector, the out global modes (GMs) must have a positive laboratory-frame group velocity.
As can be seen in Fig.8, modes no and uo have positive group velocities. So too does mode lo, except if K is negative (at very large ω). These modes are the out GMs on the left. In the low-index region on the right, the only out GM, i.e., propagat- ing away from the RIF and ahead of it, is mode moR. Thus, when calculating the laboratory frame spectral density at op- tical frequencies, we expect to detect contributions from the out GMs loL (at low ω), moR, uoL and noL.
K Ω
0Ω
ω
L(Ω
0)
ω
R(Ω
0)
Figure 3. Diagram of possible mode contributions in the laboratory frame: the optical frequency branch is shown in the laboratory frame on the left (high refractive index — orange (light grey)) and the right (low refractive index — black) of the RIF (see Fig.1). For a given laboratory frame frequency, Ω
0, modes corresponding to solutions of (A6) at two moving frame frequencies ω
R(Ω
0) and ω
L(Ω
0) may contribute to the emission. These local modes define out GMs and their contributions add up to the laboratory frame spectral density (15) and photon-number correlation (9) at Ω
0.
The moving frame flux density φ α (ω) in out GM α is (3).
The laboratory frame flux density Φ α (Ω) is obtained from this by [42]
Φ α (Ω) =
1 − u v g (Ω)
φ α (ω). (13) The total spectral density at Ω, Φ(Ω), is found by adding con- tributions of all GMs:
Φ(Ω) = X
α
Φ α (Ω). (14)
Here, we have to consider that a laboratory-frame detector might detect more than a single mode α. Assuming the de- tector is sensitive to a frequency Ω 0 , then we can read off the laboratory frame dispersion diagram (Fig.3) that there are two modes with Ω 0 and opposite K. Note that there is another pair of solutions at −Ω 0 . Out of the four solutions exactly one has positive group velocity and positive moving frame frequency ω. Thus the detector frequency corresponds to a unique mode in general, although we remark that two excep- tions are possible: (a) The detected frequency interval could contain the boundary frequency separating two modes, e.g.
moR and uoL. In this case the interval is reduced to the de- tected mode with group velocity away from the RIF. (b) Two out-GMs from either side of the RIF might share a laborat- ory frequency Ω 0 (Fig.3). In this case they typically do not share the same moving frame frequency (ω R (Ω 0 ) 6= ω L (Ω 0 )) and the covariances and variances of the modes add up, as for incoherent fields.
The wavelength equivalent to Φ(Ω) (14) is Φ λ (λ) = 2πc
λ 2 X
α
1 − u v g 2πc
λ
!
φ α (ω) , (15) with ω = γ 2πc λ − uK α
; K α the laboratory frame
wavenumber; α the modes that contribute to the emission at λ,
and φ α (ω) their moving frame flux density (3); v g the mode’s group velocity in the laboratory frame.
Equation (15) completes this section, where we formulated the important output observables for optical analogue gravity in both frames. These quantities are the tools we will use in section V to characterise and typify the emitted field in the moving and laboratory frames.
IV. ANALYTICAL CALCULATION OF THE SCATTERING MATRIX
As we have seen in the previous section, the quantum field for all modes explicitly follows from the scattering matrix, and we now build on the field theory of appendix A to derive the latter analytically for the first time. We calculate the scat-
tering matrix at all frequencies (from all kinematic scenarios) and include evanescent waves in the calculation, and thus our method lends itself to considerations of more general RIF pro- files.
We continue to use the step in the index, as in Fig.1.
For a monochromatic field of frequency ω, the canon- ical conjugate momenta Π A and Π P
ias well as their first spatial derivatives can be expressed by the electro- magnetic potential A, the polarisation fields P i as well as their derivatives in a homogeneous region using (A2), (A3) and (A5) (App. A). Therefore the field vector V ~ = (A P 1 P 2 P 3 Π A Π P
1Π P
2Π P
3) T can be reexpressed by the vector W ~ = (A, P 1 , P 2 , P 3 , A 0 , P 1 0 , P 2 0 , P 3 0 ) T . The prime de- notes the spatial derivative of the field. W ~ will allow us to simplify the matching conditions below.
V ~ and W ~ are related by Eqs.(A2)-(A5) as:
V ~ =
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
i 4πc ω
20 0 0 0 0 0 0
γ
c −i κ ωγ
1
Ω
210 0 0 − κ uγ
21
Ω
210 0
γ
c 0 −i κ ωγ
2
Ω
220 0 0 − κ uγ
22
Ω
220
γ
c 0 0 −i κ ωγ
3
Ω
230 0 0 − κ uγ
23
Ω
23
W , ~ (16)
for a field at frequency ω. We call the matrix in (16) U , and note that Det(U ) = 0. We make (16) applicable for global modes by defining U = U L θ(−x) + U R θ(x). We may com- bine eight plane wave modes α (α ∈ {α 1 , ..., α 8 }) into a mode set as a matrix W
W L/R =
W ~ L/R α
1W ~ L/R α
2... ~ W L/R α
8, (17) with W ~ α ’s ordered by laboratory-frame frequency. L/R cor- responds to a basis set of local modes on either side of the boundary at x = 0. This is similar to (A14).
In order to describe the mode coupling at the RIF with the scattering matrix formalism, we construct the global modes (GMs) of the inhomogeneous system, introduced in section A 2 of App. A. These GMs are linear combinations of the plane wave solutions W ~ α , the local modes (LMs), for each homogeneous medium on either side of the RIF. There are complete sets of global in modes,
W in (x) = W L (x) σ L in θ(−x) + W R (x) σ R in θ(x), (18) and global out modes,
W out (x) = W L (x) σ L out θ(−x) + W R (x) σ R out θ(x). (19) Each global in (out) mode contains only one LM, whose en- ergy flux is directed into (out of) the boundary. Because there are 16 LMs connected at the index boundary, there exist 16
GMs: 8 in and 8 out. In (18) and (19), each 8 × 8 σ-matrix contains the coefficients of the 8 local modes for the 8 global modes.
The mode decompositions of GMs into LMs on the left and on the right of the interface, respectively, are related by the matching conditions at the interface. For in and out modes at x = 0, we write
W in = W L σ L in = W R σ R in (20) W out = W L σ L out = W R σ R out . (21) This leaves us with 64 unknowns in either (20) or (21). The W ’s are known, so we can obtain all σ’s (we calculate an ex- ample in Appendix C). We calculate the scattering matrix by relating the in and out GMs as:
W in (x) = W out (x) S. (22) Insertion of (18) and (19) into (22) yields S:
S = σ L out −1 σ L in = σ out R −1 σ R in . (23)
Now we have demonstrated how to completely character-
ize the scattering off a RIF analytically by calculation of the
scattering matrix. In Appendix C, we exemplify one mode
configuration in detail (further calculations can be found in
[36]), and explicitly derive the in and out σ matrices, and the
corresponding scattering matrix. In Appendix D, we show
that the scattering matrix is quasi-unitary, and thus correctly
normalised.
Figure 4. Photon flux (3) of each optical mode in the moving frame for varying step heights (from bottom to top, δn = 10
−6, 2 × 10
−6(coloured lines), 10
−5, 10
−4, 10
−3, 10
−2, 10
−1): (a) noL, (b) uoL, (c) moR, (d) loL (cf. Fig.8). The vertical blue- and orange-dashed lines identify the limits of the WHI and BHI, respectively, for δn = 2 × 10
−6.
V. SPECTRA OF SPONTANEOUS EMISSION We now put the analytical formulas developed in sections II-IV into action to exemplify their use, and more importantly to study our dispersive, multimode system. The system is re- duced to the essentials of analogue horizon physics, compared to more comprehensive and realistic systems.
We calculate the scattering matrix and compute 1- and 2- photon spectra of spontaneous emission. Optical analogue experiments are different from their fluid-based counterparts (such as BECs [5], water waves [6–8] or fluids of light [12–
15]) in that the reference frames are exchanged: the rest frame of the optical experiment corresponds to the frame of the mov- ing fluid, and vice versa. In both analogues, the measurements are performed in the laboratory frame. Here, we present spec- tra and mode-correlation maps in both frames to predict ob- servations in optics and to allow comparison to fluid-based analogues.
We continue with the example of light in bulk fused silica at a step-like RIF as in section II. The Sellmeier coefficients in (1) are: κ 1,2,3 = 0.07142, 0.03246, and 0.05540 for the elastic constants, and Ω 1,2,3 = 190.341 THz, 16.2047 PHz, and 27.537 PHz for the resonance frequencies [44]. Here, we present computations for various step heights, from δn = 10 −6 to δn = 10 −1 . We first consider a RIF moving at velo- city u = 2/3c in bulk fused silica and later vary the velocity of the RIF (in section V C).
´
A. Moving frame spectra
In the moving frame, the contributions of the different modes under the various kinematic scenarios are most appar- ent. We use the formulas developed in sections II-IV to calcu- late 1- and 2-photon spectra. We also investigate the influence of the step height, which we change over a (realistic) range of 5 orders of magnitude, on the emission.
We start by computing the spontaneous photon flux (3) in the moving frame in the four optical modes, see Fig.4 ((a) noL, (b) uoL, (c) moR and (d) loL) for various step heights δn. The emission in all modes increases with step height.
The frequency intervals with horizons are located at the blue and orange dashed lines for the white hole interval (WHI) and black hole interval (BHI), respectively (kinematic scenario of Fig.2 b and d). Except for the three highest step heights, the frequency intervals with event horizons are very narrow (e.g.
1 and 10 µeV for the WHI and BHI, respectively, at δn = 2 × 10 −6 ). Here we observe the following emission peaks:
modes loL and noL both form a peak at the WHI, whereas
modes moR and noL both form a peak at the BHI. Otherwise,
modes moR and uoL sharply increase towards the WHI and
BHI, respectively, but neither of them exists there. Outside the
horizon intervals the emission is spectrally wide and smooth.
nuL
nuL
nuL
nuL
nuL
nuL nuL nuL nuL nuL noL noL noL noL
noL
noL
noL
noL
noL
noL nlL nlL nlL nlL nlL
nlL nlL nlL nlL nlL lL lL lL lL
lL
lL
lL
lL
lL
lL c/loL
c/loL
c/loL
c/loL
c/loL
c loL loL loL loL
c c c c c
c/moR
c/moR
c/moR
c/moR
c/moR c/uoL
c/uoL
c/uoL
c/uoL
c/uoL
uoL uoL uoL moR moR
uL uL uL uL uL
uL
uL
uL
uL
uL
a b c d e
C
0 0.2 0.4 0.6 0.8 1.0
Figure 5. Matrices of photon number correlations (9) between the eight outgoing modes for five typifying frequencies: columns a-d as in Fig.2 and e for high ω, and rows for values of δn (from bottom to top, 10
−6, 10
−5, 10
−4, 10
−3, 10
−2); c — a complex, unphysical mode;
uL, lL, nlL, nuL — non-optical modes (see Fig.8).
We observe that horizons increase the emission.
Spectra in the horizon intervals assume the shape of a shark fin, best seen at higher step heights (δn = 10 −3 , 10 −2 ) as the horizon intervals widen for emission into mode noL at the WHI and BHI (a), mode moR at the BHI (c) and mode loL at the WHI (d). This shape is created because on one side (large and small ω) the emission cuts off by orders of magnitude as an emitting mode (moR and loL) ceases to exist outside the interval. On the other side, the horizon condition cuts off be- cause we approach the kinematic scenario of Fig.2 c, leading to an abrupt decrease in emission. At the highest step height the BHI expands and the WHI disappears, leading to a more uniform emission. The shape of the spectrum is formed by the change in kinematic scenario [38].
Spontaneous emission occurs in pairs of photons, so we ex- pect the photon number in any mode to be correlated with that in (at least) one other mode. Partnered emission can be char-
acterised further by computing the matrix of mutual photon
number correlations between all modes. Fig.5 shows the
photon-flux correlation coefficient (9) of two modes at typify-
ing frequencies ω and for ∆ 2 = ∆ 1 ∆ 2 . Correlations (e.g. top
row) are generally strongest between the optical modes. Be-
cause noL is the unique negative-norm optical mode, we find
the strongest correlation between this mode and positive-norm
optical modes. We observe that the correlations are different
if horizons exist (Fig.5 columns b, d). Over the WHI (BHI),
there is a single large correlation between noL and loL (noL
and moR), with other correlations being small or zero. In both
cases, the pairs of modes correspond to the Hawking radiation
and the partner. Importantly, partnered emission does not oc-
cur solely in the frequency intervals where the kinematics are
analogous to the curvature of spacetime at a black- or white-
hole horizon, but occurs in the other kinematic scenarios as
well. Without horizons (Fig.5 a, c, e), significant correlations
( a )
( b ) ( c ) ( d )
Figure 6. Spectral density of emission in the laboratory frame (15). Emission from horizons of negative/positive norm: black hole 210 nm/398 nm (dashed orange in (b) and (c)) and white hole 227 nm/3.6 µm (dashed blue in (b) and (d)). There is a phase velo- city horizon at 237 nm (solid purple line in (b)).
exist between typically three mode pairs simultaneously, in- dicating multipartite entanglement. These involve non-optical modes, although the flux is weak [45]. As can be seen by comparing the columns of Fig.5, the magnitude of the cor- relation coefficients as well as their structure are surprisingly independent from the step height, although this varies over 5 orders of magnitude.
To summarise, the flux of spontaneous emission is domin- ated by white- or black hole-horizon physics and drops sig- nificantly beyond that. The spectral characteristic is a ‘shark fin’ shape. Over the analogue white- and black hole inter- vals, paired two-mode emission at optical frequencies domin- ates. These observations are robust over a wide range of step heights δn.
B. Laboratory frame spectra
Using our formulas for observables further, we now com- pute laboratory frame spectra and identify the signature ef- fects as observed in the laboratory frame. Because we have found the signatures of horizon physics to be robust against changes in the step height δn, we now limit our discussion to δn = 2 × 10 −6 .
The laboratory frame spectrum (15) is computed in Fig.6.
Similar to its moving frame counterpart, it consists of intervals of white hole, black hole and horizonless emission. Emis- sion is sharply peaked over analogue white- and black-hole intervals, and dips down at wavelengths which are velocity- matched with the RIF. These features are therefore ruled by the kinematic scenarios explored in Fig.2. The peak at 210
(227) nm corresponds to black- (white-) hole emission into the optical negative-norm mode (noL). From there, the spec- tral density decreases towards 400 nm. At 396.34 nm, mode uoL is velocity-matched to the RIF, becomes complex, and the emission drops. The drop is limited to 5 orders of mag- nitude because at 396.33 nm black hole emission into mode moR cuts on. Finally, we observe peaks at 398 nm and around 3.6 µm that account for the black hole and white hole emission into positive norm modes moR and loL, respectively. Note that, whereas in the moving frame the frequency sets the kin- ematic scenario, in the laboratory frame the spectral density may emerge from different scenarios. All peaks in Fig.6 (b), (c) and (d) exhibit the signature ‘shark fin’ structure of hori- zon emission. Clearly, these are signature effects of horizon physics in dispersive (optical) media.
The various spectral peaks and dips we have identified have narrow linewidths: about 1 nm below 250 nm, 2 nm around 400 nm and 17 nm at 3.6 µm. They are expected to have strong spectral correlations.
Fig.7 presents the photon number-correlation coefficient (9) across the spectrum of laboratory frame emission. Meas- uring these photon-number correlations convincingly reveals their vacuum fluctuations origin. Coefficients between dif- ferent modes display a single continuous contour of signi- ficant photon number correlations across the entire spectrum.
The contour indicates correlations between the negative-norm mode noL (below 237 nm) and the positive-norm modes uoL (between 237 nm and 396 nm) or moR (beyond 396 nm). The contour thus indicates where these mode pairs share a com- mon moving frame frequency ω. Note that this depends on dispersion in a nontrivial way and is not a hyperbolic re- lation. The correlation coefficients of Fig.5 and Fig.7 are mostly identical for identical frequencies ω. Along the con- tour in Fig.7, starting at 237 mn, the correlation coefficients gradually decrease as the wavelength of noL decreases and the wavelength of mode uoL, and subsequently of mode moR, increases, up to a point around (λ 1 , λ 2 )=(400 nm, 210 nm) where the black hole horizon is formed and the correlation coefficient suddenly peaks. We observe the strongest coef- ficient, 0.97, at (397 nm, 210 nm) — between modes moR and noL. Partnered emission also dominates over the WHI (3.6 µm, 227 nm) with C = 0.92 — between modes loL and noL. In both cases, the coefficients mainly deviate from unity because of weak correlations with non-optical modes (nuL and nlL). The quantum state is almost a pure two-mode squeezed vacuum [39]. The strong correlation between out- going modes only if separated by a horizon is characteristic of the Hawking effect. On the diagonal, the self-coefficients (10) characterize the photon number noise relative to Pois- son noise. These elements are very small as individual modes carry chaotic noise and the photon numbers are small.
C. Robustness of the spectra against changes in dispersion
Finally, we discuss the experimental implementation of this
optical gravity analogue via the prism of the detection of
Hawking radiation (HR). The RIF might be generated by an
200 250 300 350 400 450 200
250 300 350 400 450
200 250 300 350 400 450
Wa velength( nm )
W av el en gt h ( nm )
3.5 3.55 3.6 3.65 3.7 3.5 3.55 3.6 3.65 3.7
200 250 300 350 400 450
Wavelength( μ m )
C
0 0.2 0.4 0.6 0.8 1.0
Figure 7. Photon number correlations in the laboratory: The RIF leads to broadband entangled pair production with a near-unity coefficient at the BHI around (397 nm, 210 nm) and the WHI around (3.6 µm, 227 nm).
optical pulse via the optical Kerr effect. The velocity u then corresponds to the group velocity of that pulse. Changing u by way of dispersion, e.g. by shifting the pulse frequency, will therefore alter the wavelengths of the black- and white hole emission: these are found close to wavelengths which have the same group velocity as the RIF.
In Table I we show the laboratory-frame properties (wavelength, spectral density and spectral correlations) of white-hole and black-hole emission for four pulse group ve- locities (or wavelength λ c ): u/c = 2/3 (400 nm), 2.04/3 (800 nm), 2.05/3 (1260 nm) and 2.04/3 (1990 nm). While 1260 nm is the zero dispersion wavelength, where the group index assumes a maximum, 800 nm and 1990 nm share the same group velocity. The table shows that overall emis- sion wavelengths for the optical modes are shifted, but the correlation strengths are largely robust. The flux strength changes moderately with the pulse velocity. However, be- cause the black hole emission into mode moR is very close to the pulse wavelength, it will be difficult to distinguish it from photons of the pulse. Also, the white hole emission at infrared wavelengths is hard to detect.
When the group velocity of the RIF matches the group ve- locity of waves in the visible, the Hawking effect may create pairs of photons in the UV (the partner) and visible (HR).
This can be achieved by using a similar medium to ours, for example metamaterial waveguides, such as photonic crys- tal fibres [32, 35–37, 46]. In the anomalous dispersion re- gion, a soliton can be generated in the fibre [44]. In this case, wavelengths in the visible/UV range might be velocity matched with the pulse: the emission from the front or back of the pulse will be found in this range, where the pulse edge con- stitutes a horizon. Other nonlinear effects such as the Raman effect, phase-matched four-wave mixing or ˇ Cerenkov radi- ation may take place in the medium but may be distinguished by their respective spectral signature.
VI. CONCLUSION
We considered the mixing of modes of positive- and negative-norm in various kinematic scenarios in dispersive op- tical analogues to gravity. The refractive index front (RIF) may act as a black- or white hole horizon or as a horizonless emitter simultaneously. We presented a theoretical model (ex- panding on the canonical adaptation [42, 47] of the Hopfield model [48]), formulated observables in the relevant reference frames and calculated the two-particle spectrum of an optical analogue for the first time.
We obtained a broad and structured spectrum with emis- sion peaks in pairs of modes. The emission is in entangled photon pairs and occurs strongly from horizons on a back- ground of a broadband, weak and horizonless emission. We change the height of the RIF over 5 orders of magnitude, and find that the signatures of horizon emission are not affected:
the quantum correlation structure remains intact and the emis- sion shape does not change.
These effects are largely robust against changes in disper- sion as well, and we identify nonlinear waveguides as media of choice to observe horizon physics. The resulting spectral peak, spectral correlations and their dependence on RIF velo- city and medium dispersion are key identifiers of the type of spontaneous emission at the horizon by the Hawking effect or the horizonless emission.
This analytical method can be generalised to study other
RIF profiles, such as a finite-length pulse, and to calculate
the spectra and their spectral correlations. This is essential
to identify the optimal conditions to observe the Hawking ef-
fect. Importantly, the method can be used to provide a theory
trace for optical analogue experiments, to learn more about
the Hawking effect, and to study a variety of effects such as
cosmological pair creation by passing gravitational waves or
expanding/contracting universes [49–53], the so-called black
RIF White hole Black hole u/c λ
c(nm) λ
noL(nm) λ
loL(µm)
Φ
noL(λ) Φ
loL(λ) C λ
noL(nm) λ
moR(nm)
Φ
noL(λ) Φ
moR(λ) C
2/3 400 227 3.6 3.5 × 10
137.8 × 10
80.92 209.8 398.5 2.1 × 10
141.9 × 10
110.97 2.04/3 800 379 2.01 3.4 × 10
131.5 × 10
100.99 372 810 1.1 × 10
141.3 × 10
110.99 2.05/3 1260 438 1.36 2.4 × 10
146.1 × 10
100.99 438 1317 2.1 × 10
141.9 × 10
100.99 2.04/3 1990 379 2.01 6.7 × 10
131.5 × 10
100.99 372 810 1.2 × 10
141.4 × 10
110.99
Table I. Dependence of horizon physics on the velocity u of the RIF. λ
c: effective RIF central wavelength.
hole laser [54, 55], analogue wormholes, and the quasi-bound states of black holes [56].
ACKNOWLEDGMENTS
The authors are thankful to Bill Unruh, Joseph Cousins, Stephen Barnett, Vyome Singh and David Bermudez for in- sightful conversations. This work was funded by the EPSRC via Grant No. EP/L505079/1 and the IMPP.
[1] W. G. Unruh, Physical Review Letters 46, 1351 (1981).
[2] S. W.Hawking, Nature 248, 30 (1974).
[3] S. W. Hawking, Communications In Mathematical Physics 43, 199 (1975).
[4] O. Lahav, A. Itah, A. Blumkin, C. Gordon, S. Rinott, A. Zayats and J. Steinhauer, Physical Review Letters 105, 240401 (2010).
[5] J. R. Mu˜noz de Nova, K. Golubkov, V. I. Kolobov, and J. Stein- hauer, Nature 569, 688 (2019).
[6] G. Rousseaux, C. Mathis, P. Maissa, T. G. Philbin and U. Leon- hardt, New Journal of Physics 10, 053015 (2008).
[7] S. Weinfurtner, E. W. Tedford, M. C. J. Penrice, W. G. Un- ruh and G. A. Lawrence, Physical Review Letters 106, 021302 (2011).
[8] L.-P. Euv´e, F. Michel, R. Parentani, T. G. Philbin and G.
Rousseaux, Physical Review Letters 117, 121301 (2016).
[9] T. Torres, S. Patrick, A. Coutant, M. Richartz, E. W. Tedford and S. Weinfurtner, Nature Physics 13, 833 (2017).
[10] L.-P. Euv´e and G. Rousseaux, Physical Review D 96, 064042 (2017).
[11] L.-P. Euv´e, S. Robertson, N. James, A. Fabbri and G.
Rousseaux, Physical Review Letters 124, 141101 (2020).
[12] H. S. Nguyen, D. Gerace, I. Carusotto, D. Sanvitto, E. Galopin, A. Lemaˆıtre, I. Sagnes, J. Bloch and A. Amo, Physical Review Letters 114, 036402 (2015).
[13] C. Michel, O. Boughdad, M. Albert, P.-E. Larr´e and M. Bellec, Nature Communications 9, 2108 (2018).
[14] Q. Fontaine, T. Bienaim´e, S. Pigeon, E. Giacobino, A. Bramati and Q. Glorieux, Physical Review Letters 121 (2018).
[15] M. J. Jacquet, T. Boulier, F. Claude, A. Maˆıtre, E. Cancellieri, C. Adrados, A. Amo, S. Pigeon, Q. Glorieux, A. Bramati and E. Giacobino, arXiv:2002.00043 [cond-mat, physics:quant-ph]
(2020).
[16] X. Busch and R. Parentani, Physical Review D 89, 105024 (2014).
[17] X. Busch, I. Carusotto and R. Parentani, Physical Review A 89, 043819 (2014).
[18] S. Finazzi and I. Carusotto, Physical Review A 90, 033607 (2014).
[19] D. Boiron, A. Fabbri, P.-E. Larr´e, N. Pavloff, C. I. Westbrook and P. Zi´n, Physical Review Letters 115, 025301 (2015).
[20] F. Michel, J.-F. Coupechoux and R. Parentani, Physical Review
D 94 (2016).
[21] A. Finke, P. Jain and S. Weinfurtner, New Journal of Physics 18, 113017 (2016).
[22] R. Sch¨utzhold and W. G. Unruh, Physical Review Letters 107, 149401 (2011).
[23] A. Finke, P. Jain and S. Weinfurtner, New Journal of Physics 18, 113017 (2016).
[24] A. Fabbri and N. Pavloff, SciPost Physics 4 (2018).
[25] A. Coutant and S. Weinfurtner, Physical Review D 97, 025005 (2018).
[26] A. Coutant and S. Weinfurtner, Physical Review D 97, 025006 (2018).
[27] M. Isoard and N. Pavloff, Physical Review Letters 124, 060401 (2020).
[28] M. Visser, Classical and Quantum Gravity 15, 1767 (1998).
[29] U. Leonhardt, Nature 415, 406 (2002).
[30] R. Sch¨utzhold and W. G. Unruh, Physical Review Letters 95, 031301 (2005).
[31] A. V. Gorbach and D. V. Skyrabin, Nature Photonics 1, 653 (2007).
[32] T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. K¨onig and U. Leonhardt, Science 319, 1367 (2008).
[33] S. Hill, C. E. Kuklewicz, U. Leonhardt and F. K¨onig, Optics Express 17, 13588 (2009).
[34] D. Faccio, S. Cacciatori, V. Gorini, G. Sala, A. Averchi, A.
Lotti, M. Kolesik and J. V. Moloney, Europhysics Letters 89, 34004 (2010).
[35] A. Choudhary and F. K¨onig, Optics Express 20, 5538 (2012).
[36] M. J. Jacquet, Negative Frequency at the Horizon: Theoretical Study and Experimental Realisation of Analogue Gravity Phys- ics in Dispersive Optical Media, Springer Theses (Springer In- ternational Publishing, Vienna, 2018), ISBN 978-3-319-91070- 3.
[37] J. Drori, Y. Rosenberg, D. Bermudez, Y. Silbergerg and U.
Leonhardt, Physical Review Letters 122 (2019).
[38] M. J. Jacquet and F. K¨onig, Physical Review A 92, 023851 (2015).
[39] M. J. Jacquet and F. K¨onig, arXiv:2001.05807 [cond-mat, physics:gr-qc, physics:quant-ph] (2020).
[40] F. Belgiorno, S. Cacciatori and F. Dalla Piazza, Physical Review
D 91, 124063 (2015).
[41] M. F. Linder, R. Sch¨utzhold and W. G. Unruh, Physical Review D 93, 104010 (2016).
[42] S. Finazzi and I. Carusotto, Physical Review A 87, 023803 (2013).
[43] R. Loudon, The quantum theory of light, 117 (Oxford Univer- sity Press, Oxford, 2000).
[44] G. P. Agrawal, Nonlinear fiber optics (Academic Press, Boston, 2012).
[45] Note that also extremely weak positive-positive norm correla- tions exist which are not visible in the figure, e.g. between uoL and moR.
[46] J. McLenaghan and F. K¨onig, New Journal of Physics 16, 063017 (2014).
[47] R. Sch¨utzhold, G. Plunien and G. Soff, Physical Review Letters 88, 061101 (2002).
[48] J. J. Hopfield, Physical Review 112, 1555 (1958).
[49] S. Weinfurtner, P. Jain, M. Visser and C. W. Gardiner, Classical and Quantum Gravity 26, 065012 (2009).
[50] S. Eckel, A. Kumar, T. Jacobson, I. B. Spielman and G. K.
Campbell, Physical Review X 8 (2018).
[51] D. Campo and R. Parentani, Physical Review D 72, 045015 (2005).
[52] D. Campo and R. Parentani, International Journal of Theoretical Physics 44, 1705 (2005).
[53] D. Campo and R. Parentani, Physical Review D 74, 025001 (2006).
[54] J. L. Gaona-Reyes and D. Bermudez, Annals of Physics 380, 41 (2017).
[55] D. Bermudez and U. Leonhardt, Classical and Quantum Gravity 36, 024001 (2019).
[56] S. Patrick, A. Coutant, M. Richartz and S. Weinfurtner, Physical Review Letters 121, 061101 (2018).
[57] J. Macher and R. Parentani, Physical Review D 79, 124008 (2009).
[58] C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Photons and atoms: introduction to quantum electrodynamics, Physics textbook (Wiley, Weinheim, 2004), ISBN 978-0-471-18433-1.
[59] W. Sellmeier, Annalen der Physik 223, 386 (1872).
[60] E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D.
Twonsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. K¨onig and D. Faccio, Physical Review Letters 108, 253901 (2012).
[61] M. Schubert and B. Wilhelmi, Nonlinear optics and quantum electronics, Wiley series in pure and applied optics (Wiley, New York, 1986), ISBN 978-0-471-08807-3.
[62] B. Huttner, J. J. Baumberg and S. M. Barnett, Europhysics Let- ters 16, 177 (1991).
[63] B. Huttner and S. M. Barnett, Europhysics Letters 18, 487 (1992).
[64] R. Matloob, R. Loudon, S. M. Barnett and J. Jeffers, Physical Review A 52, 4823 (1995).
[65] S. M. Barnett, B. Huttner and R. Loudon, Physical Review Let- ters 68, 3698 (1992).
[66] S. M. Barnett, R. Matloob and R. Loudon, Journal of Modern Optics 42, 1165 (1995).
[67] D. J. Santos and R. Loudon, Physical Review A 52, 1538 (1995).
[68] S. Finazzi and I. Carosutto, Physical Review A 89, 053807 (2014).
[69] N. N. Bogoliubov, V. V. Tolmachov and D. V. Sirkov, Fortsch- ritte der Physik 6, 605 (1958).
Appendix A: Field Theory
1. Light in an inhomogeneous dispersive dielectric
Following [42, 47, 57] and [38], we describe the interac- tions of light with an inhomogeneous and transparent dielec- tric by a microscopic model based on the Hopfield model [48]. We consider one-dimensional scalar electromagnetic fields and operate at frequencies sufficiently far from the me- dium resonances to neglect absorption. The medium consists of polarisable molecules — oscillators with eigenfrequencies (resonant frequency) Ω i and elastic constants κ −1 i . Since the wavelength of light is large compared to the molecular scale, we consider the dielectric in the continuum limit and describe the electric dipole displacement by the massive scalar field P i . The electromagnetic field (a massless scalar field) is described by A, the x-component of the vector potential A(X, T ~ ) (with E ~ = −∂ T A ~ in temporal gauge), where X and T are space and time in the laboratory frame.
Our study is based on the consideration of the step-like geometry of a RIF, that propagates at constant speed u in the positive X-direction in the laboratory frame. The RIF is shown schematically in Fig.1 in the co-moving Lorentz frame coordinates x and t. We locate the boundary of the RIF at x = 0. We focus entirely on the index change induced by the RIF, neglecting phonon interactions and phase-matched optical nonlinearities such as four-wave mixing. In both ho- mogeneous regions (x ≷ 0), the interaction of the electromag- netic field with the three polarization fields of the medium is described by the Lagrangian density [38, 42, 48]
L M F = (∂ t A) 2
8πc 2 − (∂ x A) 2 8π +
N
X
i=1
γ 2 (∂ t P i − u∂ x P i ) 2 2κ i Ω 2 i
− P i 2 2κ i
+ Aγ
c (∂ t P i − u∂ x P i )
,
(A1)
where (κ i Ω 2 i ) −1 is the inertia of oscillator P i and γ = 1 − u 2 /c 2 −
1/
2. Most transparent dielectrics are sufficiently well described by three resonances and so we use N = 3. The term linear in A in Eq.(A1) describes the coupling between the fields. The Lagrangian density accounts for the free space and medium contributions to the field through the first two terms and the sum, respectively. Dispersion enters as a time depend- ence of the addends of the summation.
By the principle of least action, we obtain the Hamiltonian
density by varying the Lagrangian density (A1) with respect
to the canonical momentum densities of light and the polarisa-
tion fields. From the Hamiltonian density follow the Hamilton
equations, the equations of motion for the fields [42, 58]:
A ˙ = 4πc 2 Π A (A2)
P ˙ i = κ i Ω 2 i γ 2
Π P
i− A γ c
+ uP i
0(A3) Π ˙ A = A
004π +
3
X
i=1
κ i Ω 2 i γ 2
Π P
i− A γ c
(A4) Π ˙ P
i= − P i
κ i
+ uΠ
0P
i
, (A5)
where the derivatives are with respect to (x, t) and Π i ’s de- note the fields canonical conjugate to A and P i . We go to Fourier space by ∂ t ↔ −iω and ∂ x ↔ ik, where k and ω are, respectively, the wavenumber and frequency in the mov- ing frame. We obtain the generic Sellmeier dispersion relation of bulk transparent dielectrics [59], Eq.(1) of the main text:
c 2 k 2 = ω 2 +
3
X
i=1
4πκ i γ 2 (ω + uk) 2 1 − γ
2(ω+uk) Ω
2 2i
. (A6)
We complexify the massive field obtained from the action of (A1) by identifying harmonic plane wave solutions to (A2)- (A5) of the form
V ~ (x, t) = V ~ ¯ (ω) e ikx−iωt , (A7) where V ~ is the eight-dimensional field vector V ~ = (A P 1 P 2 P 3 Π A Π P
1Π P
2Π P
3) T . We can denote a single frequency field vector as V ~ α (ω), where α indicates a particu- lar solution k α of (A6) for ω, i.e., a ‘mode’.
By construction, the Lagrangian (A1) is invariant under global phase shifts of the dynamic fields. This continuous symmetry implies a conserved Noether current [36, 58]. As a result, the Klein-Gordon product
D V ~ 1 , ~ V 2
E
= i
~ Z
dx ~ V 1 † (x, t)
0 1 4
−1 4 0
V ~ 2 (x, t) (A8) is conserved, and so is the induced norm < ~ V , ~ V > 1/2 . Here 1 4 is the 4×4 identity matrix and the Planck constant prefactor was inserted for normalisation. It can be shown that the Klein- Gordon norm of a positive (negative) laboratory frequency Ω field is also positive (negative) [42, 46, 60]. As a result, waves of positive frequency ω in the moving frame can have either sign of the norm. This is different from fluid systems, where the sign of the norm is equal to that of the wave number K. In optical systems, the Hawking effect takes place in correlated photons of positive and negative frequency Ω, not wavenum- ber.
We orthonormalise a set of single mode field vectors V ~ α (ω) (α = α 1 ... α 8 ) using the condition [42]:
D V ~ (ω) α
1, ~ V (ω 0 ) α
2E
= sgn(Ω) δ α
1α
2δ(ω − ω 0 ). (A9) Here sgn is the sign function that determines whether the mode α 1 has positive or negative norm. Moreover, by Lorentz transform Ω = γ (ω + uk α ) and K = γ (k α + u/c 2 ω).
o o
o o
o o
o
o
